Discrete restriction estimates for manifolds avoiding a line

We identify a new way to divide the d-neighborhood of surfaces in R^3. We decompose the d-neighborhood of surfaces into a finitely-overlapping collection of rectangular boxes S. We obtain an (l^2,L^p) decoupling estimate using this decomposition, for the sharp range of exponents. The decoupling theorem we prove is new for the hyperbolic paraboloid, and recovers the Tomas-Stein restriction inequality. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line.